Why Study Math? - Solving Multi-Step Linear Inequalities - Part II
Posted: Tuesday, May 03, 2011
by Joe Pagano
Math by Joe
As we discussed in Part I, multi-step linear inequalities are quite easy to solve. Essentially, the steps involved are much like those used to solve an equation. There is one situation, however, in which the technique to solve an equation and inequality differ and that is when we multiply or divide by a negative number. The reason for this will become clear as you read on.
Solving a multi-step linear inequality such as 2x + 1 > x - 5 is quite easy and can be handled much like the solution of the equation 2x + 1 = x - 5. To keep the variable positive-so we don't have to worry about multiplying by -1-we subtract x from both sides to get x + 1 = -5. We then simply subtract 1 from the left side to isolate the variable, which gives the solution, namely x = -6. In solving the inequality, we carry out the exact same steps to get x > -6.
Now to see where equations and inequalities differ, let's take the same equation and inequality as in the last paragraph. Suppose you did not have me as a teacher and you were taught to do things the hard way. All joking aside, suppose you subtracted 2x from both sides instead of x. What happens? Well, you end up in the equation with 1 = -x -5. You then isolate the variable by adding 5 to both sides to get 6 = -x. Remember, we want to solve for x not -x. We must therefore multiply through by -1 to get -6 = x. This agrees with what we got the first time around; however, by subtracting x from both sides and keeping the variable positive, we avoided the last step of multiplying through by -1. This is always preferable.
With the inequality you will see why it is preferable to keep the variable positive. If we carry out the same steps, subtracting 2x from both sides of the inequality, we end up with 1 > -x - 5. Adding 5 to both sides, we have 6 > -x. The fatal error, or should I say trap, is when the student multiplies by -1 to get -6 > x, which is incorrect. But why?
Well there is a rule that states that when you divide or multiply an inequality by a negative number you mustreverse the direction of the inequality. So who died and made this rule a rule? Where's the rhyme or reason to this? To see the validity of the rule, let's go back to where we had 6 > -x. Instead of multiplying through by -1, let's do something else. Let's add x to both sides. Thus we get 6 + x > 0, 0 because x + -x on the right cancel to yield 0. Now subtract 6 from both sides to get x > -6. Voila. Now we see why the rule rules. Multiplying through an inequality by -1 changes the position of the variable from one side to the other, basically causing the inequality symbol to face the other way. This does not happen with equations because both the right and left sides are in balance and therefore indistinguishable.
Wow! Read the previous paragraph again because therein lies an explanation of one of the basic mysteries of mathematics. These are the things that many people go through life never having understood. And for no reason. The explanation is so easy a child could understand it. So put that in your mathematical pipe and smoke it. The aroma is sure to please, smooth and sweet. Till next time...
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